Abstract
We develop a rigorous theory of non-local Poisson structures, built on the notion of a non-local Poisson vertex algebra. As an application, we find conditions that guarantee applicability of the Lenard–Magri scheme of integrability to a pair of compatible non-local Poisson structures. We apply this scheme to several such pairs, proving thereby integrability of various evolution equations, as well as hyperbolic equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Artin, Geometric Algebra, Interscience Publishers, Inc., New York-London, 1957.
Barakat A., De Sole A., Kac V.G.: Poisson vertex algebras in the theory of Hamiltonian equations. Jpn. J. Math., 4, 141–252 (2009)
M. Błaszak, Multi-Hamiltonian theory of dynamical systems, Texts Monogr. Phys., Springer-Verlag, 1998.
Carpentier S., De Sole A., Kac V.G.: Some algebraic properties of differential operators. J. Math. Phys., 53, 063501 (2012)
S. Carpentier, A. De Sole and V.G. Kac, Rational matrix pseudodifferential operators, Selecta Math. (N.S.), to appear (2013), arXiv:1206.4165.
S. Carpentier, A. De Sole and V.G. Kac, Some remarks on non-commutative principal ideal rings, C. R. Math. Acad. Sci. Paris, 351 (2013), 5–8.
Chen H.H., Lee Y.C., Liu C.S.: Integrability of nonlinear Hamiltonian systems by inverse scattering method. Special issue on solitons in physics. Phys. Scripta, 20, 490–492 (1979)
Clarkson P.A., Cosgrove C.M.: Painlevé analysis of the nonlinear Schrödinger family of equations. J. Phys. A, 20, 2003–2024 (1987)
De Sole A., Kac V.G.: Finite vs affine W-algebras. Jpn. J. Math., 1, 137–261 (2006)
De Sole A., Kac V.G.: Lie conformal algebra cohomology and the variational complex. Comm. Math. Phys., 292, 667–719 (2009)
De Sole A., Kac V.G.: The variational Poisson cohomolgy. Jpn. J. Math., 8, 1–145 (2013)
De Sole A., Kac V.G., Wakimoto M.: On classification of Poisson vertex algebras. Transform. Groups, 15, 883–907 (2010)
L.A. Dickey, Soliton Equations and Hamiltonian Systems. Second ed., Adv. Ser. Math. Phys., 26, World Sci. Publ., 2003.
Dieudonné J.: Les déterminants sur un corps non commutatif. Bull. Soc. Math. France, 71, 27–45 (1943)
I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Nonlinear Sci. Theory Appl., John Wiley & Sons, 1993.
B.A. Dubrovin and S.P. Novikov, Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory. (Russian), Uspekhi Mat. Nauk, 44 (1989), no. 6 (270), 29–98; translation in Russian Math. Surveys, 44 (1989), no. 6, 35–124.
Fokas A.S.: On a class of physically important integrable equations. Phys. D, 87, 145–150 (1995)
E. Frenkel, Five lectures on soliton equations, In: Integrable Systems, (eds. C.-L. Terng and K.K. Uhlenbeck), Surv. Differ. Geom., 4, Int. Press, Boston, MA, 1998, pp. 131–180.
Fuchssteiner B., Fokas A.S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D, 4, 47–66 (1981)
I.M. Gelfand and I. Dorfman, Schouten bracket and Hamiltonian operators, Funktsional. Anal. i Prilozhen., 14 (1980), no. 3, 71–74.
Kaup D.J., Newell A.C.: An exact solution for a derivative nonlinear Schrödinger equation. J. Mathematical Phys., 19, 798–801 (1978)
Magri F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys., 19, 1156–1162 (1978)
F. Magri, A geometrical approach to the nonlinear solvable equations, In: Nonlinear Evolution Equations and Dynamical Systems, Lecture Notes in Phys., 120, Springer-Verlag, 1980, pp. 233–263.
Maltsev A.Ya., Novikov S.P.: On the local systems Hamiltonian in the weakly non-local Poisson brackets. Phys. D, 156, 53–80 (2001)
A.G. Meshkov and V.V. Sokolov, Integrable evolution equations with constant separant, Ufa Mathematical Journal, 4 (2012), no. 3, 104–153.
A.V. Mikhaĭlov, A.B. Shabat and V.V. Sokolov, The symmetry approach to the classification of integrable equations, In: Integrability and Kinetic Equations for Solitons, Naukova Dumka, Kiev, 1990, pp. 213–279.
P.J. Olver, Applications of Lie Groups to Differential Equations. Second ed., Grad. Texts in Math., 107, Springer-Verlag, 1993.
Sakovich A., Sakovich S.: The short pulse equation is integrable. J. Phys. Soc. Japan, 74, 239–241 (2005)
Sanders J.A., Wang J.P.: On recursion operators. Phys. D, 149, 1–10 (2001)
Schäfer T., Wayne C.E.: Propagation of ultra-short optical pulses in cubic nonlinear media. Phys. D, 196, 90–105 (2004)
V.V. Sokolov, Hamiltonian property of the Krichever–Novikov equation. (Russian), Dokl. Akad. Nauk SSSR, 277 (1984), no. 1, 48–50.
L.A. Takhtadzhyan and L.D. Faddeev, The Hamiltonian Approach in Soliton Theory, Nauka, Moscow, 1986.
D. Talati and R. Turhan, On a recently introduced fifth-order bi-Hamiltonian equation and trivially related Hamiltonian operators, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 081.
Wadati M., Konno K., Ichikawa Y.H.: A generalization of inverse scattering method. J. Phys. Soc. Japan, 46, 1965–1966 (1979)
Wang J.P.: Lenard scheme for two-dimensional periodic Volterra chain. J. Math. Phys., 50, 023506 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Yasuyuki Kawahigashi
Dedicated to Minoru Wakimoto on his 70th birthday
The first author is supported in part by Department of Mathematics, MIT.
The second author is supported in part by an NSF grant, and the Simons Fellowship.
About this article
Cite this article
De Sole, A., Kac, V.G. Non-local Poisson structures and applications to the theory of integrable systems. Jpn. J. Math. 8, 233–347 (2013). https://doi.org/10.1007/s11537-013-1306-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11537-013-1306-z
Keywords and phrases
- non-local Poisson vertex algebra
- non-local Poisson structure
- rational matrix pseudo-differential operators
- Lenard–Magri scheme of integrability
- bi-Hamiltonian integrable hierarchies